Hearst index - page 27
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To avoid confusion, let us refer to the definition of ME: the mathematical expectation is the average of a series of returns of a random variable.
The ME of a sample = the average of the sample. And the type of series from which the sample is drawn is irrelevant to the definition of ME. But not the point.
For everything else, we understand each other.
The distribution is normal, with zero MO and a given standard deviation. In this context, consistency and trendiness are the same thing. When I say "trend series" it means that the probability of coincidence of the sign of the increment with the sign of its previous returns is above 50%, anti-trendicity is the opposite, the probability of coincidence of the sign is less than 50%. This is not my definition, but exactly what is meant in the book.
Despite the public's tepid interest in the stated theme, I am continuing my following of Peters' book.
The entire Peters series is then divided into independent sub-periods. Each sub-period is calculated according to the above methodology. As a result, there is some average RS value, and it must be qualitatively different from Brownian motion. Since the dispersion of particles will be directly proportional to the logarithm of the period, the Hurst ratio, i.e. the ratio of timespan to period, must be a constant and be 0.5. In fact the formula is not perfect and tends to overestimate the result by 0.3, i.e. on obviously random series, Hurst will show 0.53, rather than 0.50. And it is not caused by the small sample, the more data we use, the more accurate indicator will be in the 0.53 range.
....
As you can see there are two main problems with the indicator: at sharp reversals the MO will be insignificant, while the swing will be high, which leads to the unreasonable overstatement of the indicator. On the contrary, in a clear uptrend the MO will be the main portion of the movement, but fluctuations around the MO will be small and thus the heurst will be again lower than it should be.
Thus, we can draw a preliminary conclusion that the suggested method can not adequately describe the market price movement and effectively identify trend and anti-trend components.
The reason is that volatility and, respectively, the cascade used in the formula does not converge to a constant. In this case, for the frequency of an expiration we need to divide it into "independent subperiods", so that the skew converges to the constant. I.e. do not take them out of the blue.
But all the same, it is pointless to take the series as a whole and check it for consistency. The hospital average will insignificantly differ from SB because sometimes the series is trending and sometimes it is flat. We should know when it is trending and when it is flat and why. We need to know when it is trending and when it is flat.)
There is another important point, which is not considered when using Hearst on "financial series". The point is that there is a significant "similarity" between the dynamics of the Nile floods and Hirst's experiments with a pack of cards, but not with "financial series".
Could you expand on the answer in more detail? Every year, the Nile's flooding varies in a certain range. This is his return series. It is clear that the flood will always be a positive value, so we need to detrend this series relative to its MO. Then we look at the accumulated series: the maxima and minima will form the spread. If the spill each year is random and independent, then the resulting series will be random and move along a bell-shaped trajectory with respect to time. If the series is not random and persistent, it will tend to go beyond the conditional bell-shaped trajectory; if it is antitrendous, it will be deep inside the bell.
The main problem here is slightly different. This method will work well when the expectation is more or less stable, like in the case of the Nile or solar activity. But it doesn't work with markets, and it has a different MO at each point in time. We can't deduct the MO from the market series in this case because we don't know if it is part of the spread or the stationary component of the process. More "advanced" techniques such as linear regression won't work either, because also the trend (regression line) is non-stationary, and therefore it may be the result of a deterministic process.
The reason is that the volatility and thus the sco used in the formula does not converge to a constant. It is necessary to divide the frequency of an equilibrium into "independent sub-periods", so that the sko converges to the constant. I.e. do not take them at random.
Volatility is only a measure of normalisation. The period spread is divided by its s.c.o. only to obtain the same scale for all possible series. Additionally the s.q.o. for a finite period is a finite value. It will not coincide with adjacent periods, but for its period will be single-valued, and therefore in relation to the obtained range of this period will be quite adequate value of normalization.
This is why I have specifically made the calculation for independent sub-periods. That is, if the series consists of 1000 values, and the averaging period is 100, then take 10 consecutive sub-periods of 100 values, for each of them calculate its RS, and then derive the average of these RS.
But it is still pointless to take the series as a whole and check it for consistency. The hospital average will slightly differ from the RS because sometimes the series is trending and sometimes it is flat. We should know when it is trending and when it is flat and why. We need to know when it is trending and when it is flat.)
I've been thinking about this as well. I specifically wrote a Hearst sliding indicator for this, which calculates its value at each moment in time. I haven't managed to find any qualitative patterns. But it has plenty of disadvantages, for example Hearst will overestimate its values at price reversals and underestimate them in a strong trend.
Volatility is only a measure of normalisation. We divide the range of a period by its s.c.o. only to get one scale for all possible series. Additionally the s.q.o. for a finite period is a finite value. It will not coincide with the adjacent periods, but for its period will be single-valued and therefore in relation to the obtained range of this period will be quite an adequate value of normalization.
That's why I specifically made the calculation for independent sub-periods. That is, if the series consists of 1000 values, and the averaging period is 100, then 10 successive sub-periods of 100 values are taken, for each of them a different RS is calculated, and then the average value of these RS is derived.
Of course, we will get a certain value of sko at a particular period, but that does not mean that the volatility on it will converge to a constant. Volatility in real financial series is volatile and is not characterised by a single number. Therefore "sub-periods" may contain pieces of high and low volatility and the formula will not read correctly. For example, we took a sub-period equal to one day from 0h to 24h. Volatility at different times of the day is stably different, by several times. The average value does not characterize the whole period and the Hurst calculated on its basis and taking the period into account will show who knows what. The whole Hurst formula is based on the fact that the ox will not be steadily variable in sub-periods, but will be characterized by the average value.
Could you expand on the answer in more detail? Every year, the Nile's flooding varies over a certain range. This is his return series. It is clear that the flood will always be a positive value, so we need to detrend this series relative to its MO. Then we look at the accumulated series: the maxima and minima will form the spread. If the spill each year is random and independent, then the resulting series will be random and move along a bell-shaped trajectory with respect to time. If the series is not random and persistent, it will move out of the conditional bell-shaped trajectory more often; if it is an entrendous series, it will be deep inside the bell.
The minima, maxima, spreads, etc. - it is all clear. The point is about something else.
Hurst tested it on a pack of cards to show that his method works in principle. There was a tricky arrangement of cards, which one is not important. The main thing is that his experiments clearly defined what an elementary event is.
For the Nile, as far as I remember, he also defined such elementary event, the maximum mark of the water level rise in a year (or he had the flow rate there - I don't remember). No other, intermediate values were considered. It is clear that the "physics" of the process is always constant. How much water collected in the Nile basin, how much flowed out through the channel. Basically, if it were a barrel, there would be nothing, but the Nile basin has a certain inertia (the scale of several years) in water collection/release, and that is what forms the "memory". It is important to understand that the same thing happens every year, in a certain season, water is collected from the atmosphere in a huge basin, slowly seeps through the soils into the Nile and flows towards the sea.
Now, if we calculate the Hurst coefficient for the Nile, we break down a series of these elementary homogeneous events into a series, over which we perform mathematical manipulation.
Imagine that the elementary event would be a level measurement per month, every first day. We simply took, and declared that now the elementary event would not be as it happens in nature, but as we like. So, we take those months, those that are the rainy season and those that are the drought, and break them up into a series. And so on. The result, in my opinion, is well predictable.
That's, that's my opinion of it all.
The problem with financial series is exactly the same, there is no elementary event that characterizes the process. More precisely, a notional slicing into bars is not an event in my opinion. What do I care if last minute Vasya was buying and moving the price by a few pips, and John was selling the next minute. It's like water droplets seeping into the Nile. I wonder what's going on in the aggregate.
ZS. by the way, the ideas of looking for accumulation\distribution, Wyckoff etc. - it's just from understanding that elementary events in the market are not bars at all.
For those who don't understand what it's all about, statistical operations can only be performed on elementary events.
The main problem here is seen to be somewhat different. The method will work well when the mathematical expectation (the basis, what we calculate) is more or less stable, like in the case of the Nile or solar activity. But it doesn't work with markets, and it has a different MO at each point in time. We can't subtract the MO from the market series in this case because we don't know if it is part of the spread or the stationary component of the process. More "advanced" techniques like linear regression will not work either, because also the trend (regression line) is non-stationary, and therefore it may be the result of a deterministic process.
And I would also add (as I myself have also calculated in Hearst's Excel), that the prognostic property of these statistics is doubtful. Yes, we know that the market was so and so but who knows what it will be in the next 100-1000 bars? What do you think?
Matroskin's problems were due to his lack of intelligence, while we all have problems from his excess and over-education.
Let's leave the Nile and its millennia-old history alone and come down to earth.
We have the far right bar and we are interested in the forecast for the next bar. If we take into account that it may be M1, H1 or D1, the horizon problem is solved.
Now let's answer the question: how many preceding bars are needed for forecasting the next one. I once read that t-statistics changes into z-statistics when the number of observations is more than 30. Let's triple it and get 100. For H1 there are 118 observations in a week. Most likely a new week on H1 will give new problems. That's it.
Now we make a one-step forecast. For example, we draw a straight line on the last 3 points and extend forward.
Now. Let's admit that this prediction is represented by a random variable. It follows that there is an error in calculation of this forecast. And this error is the root of the matter. If it has mo and volu at least approximately a constant, it is one thing. Or if it is not great and it can be replaced with a spread, that's also nothing. But the problem is the error.
And God forbid the prediction error looks like this.
And now we face the task of getting the stationary characteristics of the error from our limited sample.
I think so.